A subgroup H of a finite group G is said to be an <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-subgroup of G if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$N_{G}(H)\cap H^{g}\le H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mi>H</mi> <mi>g</mi> </msup> <mo>≤</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$g\in G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>; and H is said to be a weakly <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-embedded in G, if there exists a normal subgroup K of G such that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$H^{G}=HK$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>G</mi> </msup> <mo>=</mo> <mi>H</mi> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$N_{G}(H\cap K)\cap (H\cap K)^{g}\le H\cap K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo>∩</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo>∩</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mi>g</mi> </msup> <mo>≤</mo> <mi>H</mi> <mo>∩</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation>, for all <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$g\in G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we continue the research on weakly <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-embedded subgroups introduced by Asaad and Ramadan (2016) [On weakly <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-embedded subgroups of finite groups. Commun. Algebra 44:4564-4574].

A note on weakly $\mathcal {H}$-embedded subgroups of finite groups

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2026年7月13日

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Muhammad Tanveer Hussain

摘要

A subgroup H of a finite group G is said to be an $\mathcal {H}$ $H$ -subgroup of G if $N_{G}(H)\cap H^{g}\le H$ $N_{G} (H) \cap H^{g} \leq H$ for all $g\in G$ $g \in G$ ; and H is said to be a weakly $\mathcal {H}$ $H$ -embedded in G, if there exists a normal subgroup K of G such that $H^{G}=HK$ $H^{G} = H K$ and $N_{G}(H\cap K)\cap (H\cap K)^{g}\le H\cap K$ $N_{G} (H \cap K) \cap {(H \cap K)}^{g} \leq H \cap K$ , for all $g\in G$ $g \in G$ . In this paper, we continue the research on weakly $\mathcal {H}$ $H$ -embedded subgroups introduced by Asaad and Ramadan (2016) [On weakly $\mathcal {H}$ $H$ -embedded subgroups of finite groups. Commun. Algebra 44:4564-4574].

A note on weakly \(\mathcal {H}\)-embedded subgroups of finite groups

摘要